Gnomonic cubes: a simultaneous view of the extended graphic beta move

Recall that the extended  beta move is equivalent with the pair  of  moves :

beta_beta_star

where the first move is the graphic beta move and the second move is the dual of the beta move, where duality is (still loosely) defined by the following diagram:

correspondence_1

In this post I want to show you that it is possible to view simultaneously these two moves. For  that I need to introduce the gnomonic cube. (Gnomons appeared several times in this blog, in expected or unexpected places, consult the dedicated tag “gnomon“).

From the wiki page about the gnomon,     we see that

A three dimensional gnomon is commonly used in CAD and computer graphics as an aid to positioning objects in the virtual world. By convention, the X axis direction is colored red, the Y axis green and the Z axis blue.

3DGraphicsGnomon

(image taken from the mentioned wiki page, then converted to jpg)

A gnomonic cube is then just a cube with colored faces. I shall color the faces of the gnomonic cube with symbols of the gates from graphic lambda calculus! Here is the construction:

gnomonic_cube_2

So, to each gate is associated a color, for drawing conveniences. In the upper part of the picture is described how the faces of the cube are decorated. (Notice the double appearance of the \Upsilon gate, the one used as a FAN-OUT.)  In the lower part of the picture are given 4 different views of the gnomonic cube. Each face of the cube is associated with a color. Each color is associated with a gate.

Here comes the simultaneous view of the pair of moves which form, together, the extended beta move.

gnomonic_cube_3

In this picture is described a kind of a 3D move, namely the pair of gnomonic cubes connected with the blue lines can be replaced by the pair of red lines, and conversely.

If you project to the UP face of the dotted big cube then you get the graphic beta move. The UP view is the viewpoint from lambda calculus (metaphorically speaking).

If you project to the RIGHT face then you get the dual of the graphic beta move. The RIGHT view is  the viewpoint from emergent algebras (…).

Instead of 4 gates (or 5 if we count \varepsilon^{-1} as different than \varepsilon), there is only one: the gnomonic cube. Nice!

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